4. Draw a sketch graph of \( f(x)=\left(\frac{1}{3}\right)^{x} \). 4.1 Give the domain and range of \( f \). 4.2 For which values of \( x \) is \( f(x)>0 \) ? 4.3 Give the coordinates of any three points on \( f \). 4.4 Give the equation of \( g \) if \( g \) is formed when the graph of \( f \) is moved 3 units down. 4.5 Give the equation of \( h \) if \( h \) is formed when the graph of \( f \) is moved 3 units left. 4.6 Give the equation of \( k \) if \( k \) is formed when the graph of \( f \) is moved 2 units upwards and 3 un to the right. 4.7 Give the domain and range of \( g \). 4.8 Give the domain and range of \( h \). 4.9 Give the domain and range of \( k \). 4.10 Give the asymptotes of \( f, g, h \) and \( k \).

Function by following steps: - step0: Find the y-intercept: \(g\left(x\right)=3^{-x}-3\) - step1: Set \(x\)=0\(:\) \(g\left(0\right)=3^{-0}-3\) - step2: Evaluate: \(g\left(0\right)=3^{0}-3\) - step3: Evaluate the power: \(g\left(0\right)=1-3\) - step4: Subtract the numbers: \(g\left(0\right)=-2\) Analyze the y intercept of the function \( k(x)=(\frac{1}{3})^{x-3}+2 \) Function by following steps: - step0: Find the y-intercept: \(k\left(x\right)=3^{-x+3}+2\) - step1: Set \(x\)=0\(:\) \(k\left(0\right)=3^{-0+3}+2\) - step2: Evaluate: \(k\left(0\right)=3^{0+3}+2\) - step3: Simplify: \(k\left(0\right)=29\) Analyze the y intercept of the function \( f(x)=(\frac{1}{3})^{x} \) Function by following steps: - step0: Find the y-intercept: \(f\left(x\right)=\left(\frac{1}{3}\right)^{x}\) - step1: Set \(x\)=0\(:\) \(f\left(0\right)=\left(\frac{1}{3}\right)^{0}\) - step2: Evaluate the power: \(f\left(0\right)=1\) Analyze the y intercept of the function \( h(x)=(\frac{1}{3})^{x+3} \) Function by following steps: - step0: Find the y-intercept: \(h\left(x\right)=\left(\frac{1}{3}\right)^{x+3}\) - step1: Set \(x\)=0\(:\) \(h\left(0\right)=\left(\frac{1}{3}\right)^{0+3}\) - step2: Remove 0: \(h\left(0\right)=\left(\frac{1}{3}\right)^{3}\) - step3: Simplify: \(h\left(0\right)=\frac{1}{27}\) Find the horizontal asymptotes of \( f(x)=(\frac{1}{3})^{x} \). Function by following steps: - step0: Find the horizontal asymptotes: \(f\left(x\right)=\left(\frac{1}{3}\right)^{x}\) - step1: Find the horizontal asymptotes: \(\textrm{No horizontal asymptotes}\) Find the horizontal asymptotes of \( k(x)=(\frac{1}{3})^{x-3}+2 \). Function by following steps: - step0: Find the horizontal asymptotes: \(k\left(x\right)=3^{-x+3}+2\) - step1: Find the horizontal asymptotes: \(\textrm{No horizontal asymptotes}\) Find the horizontal asymptotes of \( h(x)=(\frac{1}{3})^{x+3} \). Function by following steps: - step0: Find the horizontal asymptotes: \(h\left(x\right)=\left(\frac{1}{3}\right)^{x+3}\) - step1: Find the horizontal asymptotes: \(\textrm{No horizontal asymptotes}\) Find the horizontal asymptotes of \( g(x)=(\frac{1}{3})^{x}-3 \). Function by following steps: - step0: Find the horizontal asymptotes: \(g\left(x\right)=3^{-x}-3\) - step1: Find the horizontal asymptotes: \(\textrm{No horizontal asymptotes}\) Let's solve the problem step by step. ### 4. Sketch Graph of \( f(x) = \left(\frac{1}{3}\right)^{x} \) 1. **Domain and Range of \( f \)**: - The function \( f(x) = \left(\frac{1}{3}\right)^{x} \) is defined for all real numbers \( x \). - As \( x \) approaches infinity, \( f(x) \) approaches 0 but never reaches it. As \( x \) approaches negative infinity, \( f(x) \) approaches infinity. - Therefore, the domain is \( (-\infty, \infty) \) and the range is \( (0, \infty) \). 2. **Values of \( x \) for which \( f(x) > 0 \)**: - Since \( f(x) = \left(\frac{1}{3}\right)^{x} \) is an exponential function, it is always positive for all real \( x \). - Thus, \( f(x) > 0 \) for all \( x \in (-\infty, \infty) \). 3. **Coordinates of Three Points on \( f \)**: - Let's calculate three points: - \( f(0) = \left(\frac{1}{3}\right)^{0} = 1 \) → Point: \( (0, 1) \) - \( f(1) = \left(\frac{1}{3}\right)^{1} = \frac{1}{3} \) → Point: \( (1, \frac{1}{3}) \) - \( f(-1) = \left(\frac{1}{3}\right)^{-1} = 3 \) → Point: \( (-1, 3) \) 4. **Equation of \( g \)** (moved 3 units down): - The equation of \( g \) is \( g(x) = f(x) - 3 = \left(\frac{1}{3}\right)^{x} - 3 \). 5. **Equation of \( h \)** (moved 3 units left): - The equation of \( h \) is \( h(x) = f(x + 3) = \left(\frac{1}{3}\right)^{x + 3} = \left(\frac{1}{3}\right)^{x} \cdot \left(\frac{1}{3}\right)^{3} = \frac{1}{27} \left(\frac{1}{3}\right)^{x} \). 6. **Equation of \( k \)** (moved 2 units upwards and 3 units to the right): - The equation of \( k \) is \( k(x) = f(x - 3) + 2 = \left(\frac{1}{3}\right)^{x - 3} + 2 \). 7. **Domain and Range of \( g \)**: - The domain of \( g \) is the same as \( f \): \( (-\infty, \infty) \). - The range of \( g \) is \( (-3, \infty) \) since it shifts down by 3. 8. **Domain and Range of \( h \)**: - The domain of \( h \) is \( (-\infty, \infty) \). - The range of \( h \) is \( (0, \frac{1}{27}) \) since it is a scaled version of \( f \). 9. **Domain and Range of \( k \)**: - The domain of \( k \) is \( (-\infty, \infty) \). - The range of \( k \) is \( (2, \infty) \) since it shifts up by 2. 10. **Asymptotes of \( f, g, h, \) and \( k \)**: - All functions have a horizontal asymptote at \( y = 0 \) since they approach 0 as \( x \) approaches infinity. ### Summary of Results: - **Domain of \( f \)**: \( (-\infty, \infty) \) - **Range of \( f \)**: \( (0, \infty) \) - **Values of \( x \) for \( f(x) > 0 \)**: All \( x \in (-\infty, \infty) \) - **Three Points on \( f \)**: \( (0, 1), (1, \frac{1}{3}), (-1, 3) \) - **Equation of \( g \)**: \( g(x) = \left(\frac{1}{3}\right)^{x} - 3 \) - **Equation of \( h \)**: \( h(x) = \left(\frac{1}{3}\right)^{x + 3} \) - **Equation of \( k \)**: \( k(x) = \left(\frac{1}{3}\right)^{x - 3} + 2 \) - **Domain of \( g \)**: \( (-\infty, \infty) \) - **Range of \( g \)**: \( (-3, \infty) \) - **Domain of \( h \)**: \( (-\infty, \infty) \) - **Range of \( h \)**: \( (0, \frac{1}{27}) \) - **Domain of \( k \)**: \( (-\infty, \infty) \) - **Range of \( k \)**: \( (2, \infty) \) - **Asymptotes**: All functions have a horizontal asymptote at \( y = 0 \).